Aim for FivePhysics C: E&M

Translation between representations (TBR)

Magnetic field of a wire with a nonuniform current

  • Unit 12
  • 12 points
  • About 28 minutes

You can use a calculator on this question, just like on exam day.

A multipart problem about one situation shown in several ways. You draw a diagram, derive equations, sketch or draw graphs, and then explain whether your answers agree with each other or use them to predict what happens when the situation changes. On the exam: Question 2 of 4. Section II has 4 questions in 95 minutes (50% of the exam score), one of each type in this order; calculator and equation sheet allowed. The CED suggests 25–30 minutes.

The question

A long, straight, solid cylindrical conductor of radius R carries a steady current directed out of the page along its axis. The current density is not uniform: its magnitude is J(r)=J0rRJ(r) = J_0\dfrac{r}{R}, where r is the distance from the axis and J₀ is a positive constant.

Suggested time: 28 minutes

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Part (a)

2 points

Derive an expression for the total current I in the conductor, in terms of J₀ and R.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (b)

3 points

Use Ampère’s law to derive expressions for the magnitude of the magnetic field (i) inside the conductor (r < R) and (ii) outside the conductor (r > R).

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (c)

3 points

You can’t draw here, so describe the graph you would sketch of the magnetic field magnitude B as a function of r, from r = 0 to r = 3R.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (d)

1 point

Describe the magnetic field lines inside and around the conductor, as seen looking at its cross section with the current coming out of the page.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (e)

3 points

A second conductor has the same radius R and the same total current I, but its current density is uniform. (i) At r = R/2, is the magnetic field of the second conductor greater than, less than, or equal to that of the first? (ii) By what factor? (iii) How do the two fields compare outside the conductors (r > R)? Justify your answers.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

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